Linear seismic inversion
Inverse modeling is a mathematical technique where the objective is to determine the physical properties of the subsurface of an earth region that has produced a given seismogram.
Cooke and Schneider (1983)[1] defined it as calculation of the earth's structure and physical parameters from some set of observed seismic data.
The underlying assumption in this method is that the collected seismic data are from an earth structure that matches the cross-section computed from the inversion algorithm.
[2] Some common earth properties that are inverted for include acoustic velocity, formation and fluid densities, acoustic impedance, Poisson's ratio, formation compressibility, shear rigidity, porosity, and fluid saturation.
The method has long been useful for geophysicists and can be categorized into two broad types:[3] Deterministic and stochastic inversion.
As such, this method of inversion to which linear inversion falls under is posed as an minimization problem and the accepted earth model is the set of model parameters that minimizes the objective function in producing a numerical seismogram which best compares with collected field seismic data.
On the other hand, stochastic inversion methods are used to generate constrained models as used in reservoir flow simulation, using geostatistical tools like kriging.
The aim of the technique is to minimize a function which is dependent on the difference between the convolution of the forward model with a source wavelet and the field collected seismic trace.
In the following subsections we will describe in more detail, in the context of linear inversion as a minimization problem, the different components that are necessary to invert seismic data.
[1] According to Wiggins (1972),[4] it provides a functional (computational) relationship between the model parameters and calculated values for the observed traces.
Depending on the seismic data collected, this model may vary from the classical wave equations for predicting particle displacement or fluid pressure for sound wave propagation through rock or fluids, to some variants of these classical equations.
For example, the forward model in Tarantola (1984)[5] is the wave equation for pressure variation in a liquid media during seismic wave propagation while by assuming constant velocity layers with plane interfaces, Kanasewich and Chiu (1985)[6] used the brachistotrone model of John Bernoulli for travel time of a ray along a path.
In Cooke and Schneider (1983),[1] the model is a synthetic trace generation algorithm expressed as in Eqn.
Thus, the forward model by Cooke and Schneider (1983)[1] can only be used to invert CMP data since the model invariably assumes no spreading loss by mimicking the response of a laterally homogeneous earth to a plane-wave source where s(t) = synthetic trace, w(t) = source wavelet, and R(t) = reflectivity function.
Irrespective of the definition used, numerical solution of the inverse problem is obtained as earth model that minimize the objective function.
These constraints, according to Francis 2006,[3] help to reduce non-uniqueness of the inversion solution by providing a priori information that is not contained in the inverted data while Cooke and Schneider (1983)[1] reports their useful in controlling noise and when working in a geophysically well-known area.
Generally, the numerically generated seismic data are non-linear functions of the earth model parameters.
and its comments are such that each column is the partial derivative of a component of the forward function with respect to one of the unknown earth model parameters.
Similarly, each row is the partial derivative of a component of the numerically computed seismic trace with respect to all unknown model parameters.
, is solved for as shown below, using any of the classical direct or iterative solvers for solution of a set of linear equations.
Considering that inverse modeling problem is only theoretically solvable when the number of discrete intervals for sampling the properties is equal to the number of observation in the trace to be inverted, a high-resolution sampling will lead to a large matrix which will be very expensive to invert.
Furthermore, the matrix may be singular for dependent equations, the inversion can be unstable in the presence of noise and the system may be under-constrained if parameters other than the primary variables inverted for, are desired.
In relation to parameters desired, other than impedance, Cooke and Schneider (1983)[1] gives them to include source wavelet and scale factor.
Although this example does not directly relate to seismic inversion since no traveling acoustic waves are involved, it nonetheless introduces practical application of the inversion technique in a manner easy to comprehend, before moving on to seismic applications.
that minimizes the difference between the observed temperature distribution and those obtained using the forward model of Eqn.
Source:[8] This examples inverts for earth layer velocity from recorded seismic wave travel times.
5 shows the initial velocity guesses and the travel times recorded from the field, while Fig.
6a shows the inverted heterogeneous velocity model, which is the solution of the inversion algorithm obtained after 30 iterations.
This example, taken from Cooke and Schneider (1983),[1] shows inversion of a CMP seismic trace for earth model impedance (product of density and velocity) profile.
Also recorded alongside the seismic trace is an impedance log of the earth region as shown in Fig.
